Linear norm

This is the wiki page for understanding seminorms of linear growth on a group [math]\displaystyle{ G }[/math] (such as the free group on two generators). These are functions [math]\displaystyle{ \| \|: G \to [0,+\infty) }[/math] that obey the triangle inequality

[math]\displaystyle{ \|xy\| \leq \|x\| + \|y\| }[/math]

and the linear growth condition

[math]\displaystyle{ \|x^n \| = |n| \|x\| }[/math]

for all [math]\displaystyle{ x,y \in G }[/math] and [math]\displaystyle{ n \in {\bb Z} }[/math].

Threads

• https://terrytao.wordpress.com/2017/12/16/bi-invariant-metrics-of-linear-growth-on-the-free-group/, Dec 16 2017.
• Bi-invariant metrics of linear growth on the free group, II, Dec 19 2017.